Quantification and Prediction of Extreme Events in Stochastic Systems

Rogue or freak waves are ocean waves whose height is abnormally large for a particular sea state. Often described an enormous "wall of water," (e.g. the New Year wave in Fig. 1) such waves have caused catastrophic damage to ships and coastal structures. For example, in 1978 the German super-tanker München vanished, along with her 26 crew members. Searches for the ship recovered little, but a lifeboat was recovered whose attachment pins showed evidence of being subjected to a great force. As this lifeboat was stowed 20m above the water line, some have conjectured that the München may have been struck by an extremely large wave [Liu, Geofizika 24, 2007]. Here we describe our method for reliably predicting these rogue waves before they occur while expending minimal computational effort, which we term Reduced Order Prediction of Extremes (ROPE).

The large, steep nature of these rogue waves, combined with recent evidence that they can occur more likely than Gaussian statistics would suggest, imply that nonlinear models are necessary to fully understand their dynamics. Thus, we focus our attention on models that incorporate this nonlinearity while remaining simple enough to be tractable. Two such examples are the equation of Majda, McLaughlin, and Tabak (MMT) and the Nonlinear Schrodinger Equation (NLS). The MMT model is similar to NLS, differing in that it includes a "fractional" differential operator in place of the 2nd derivative in NLS [Majda et. al., J. Nonlinear Sci. 6, 1997]. Although NLS and MMT are quite simple, they have attracted attention due to an inherent mechanism for generating rogue waves. Specifically, simple periodic solutions to both equations are unstable, with small perturbations initiating "focusing" events, where the nearby wave field is soaked up to produce a large, localized wave. This process, known as the Benjamin-Feir instability, is well understood theoretically and has been reproduced in experiment [Chabchoub et. al. PRL 106, 2011].

Figure 1: The New Year wave recorded at the Draupner platform in the North Sea on January 1, 1995.

Of course, in realistic settings the ocean surface is not a near plane wave. We are interested in developing a scheme to predict rogue waves arising in realistic ocean spectra where energy is distributed over a range of scales. One approach for doing so would be to numerically solve the fully nonlinear water wave equations directly. However, such an approach would require an immense computational effort. Instead, we have developed a simple method to reliably characterize the "trigger" of these rogue wave events. By projecting the field onto a carefully selected basis, we are able to accurately able to predict rogue waves before they occur for minimal computational expense.

We now describe our method for Reduced Order Prediction of Extremes (ROPE). The key to our approach is the following observation: certain localized wave groups initiate dramatic focusing of wave energy, creating a rogue wave (see figure 2 below). We have performed a detailed analysis of which types of wave groups are likely to trigger rogue wave formation, and we identify such groups by projecting the field onto a carefully tuned Gabor wavelet basis [Cousins, Sapsis Physica D, 2014]. Although energy is spread over roughly 8000 modes in our simulations of MMT, we are able to effectively characterize the initiation of rogue waves by considering roughly 100 modes. The reason for this is that we have found that there is a particular length scale at which energy localization is likely to induce a rogue wave.

We then compute the quantity Y0, which is the amount of energy localized at this critical scale. If we find that Y0 is large, then we expect that an upcoming rogue wave is likely. In fact, there is a critical value of Y0 which, when exceeded, implies that an extreme event is almost guaranteed to occur. This allows the ROPE method to provide predictive utility—for a given wave field, we compute Y0 and translate this into a probability of an upcoming extreme event.

Figure 3: Family of conditional densities for upcoming wave elevation given current value of Y0 (left). Right, probability of future rogue wave (EE) given current value of Y0.

The ROPE approach has demonstrated impressive predictive skill for the MMT and NLS models. For example, in tests on 50 simulations of the MMT model we predicted that a rogue wave would occur whenever Y0 exceeded 1.1 (which, according to the above figure, means that the probability of an upcoming rogue wave should be roughly 0.8). Of 191 rogue wave predictions, 155 correctly predicted a rogue wave, meaning that the false positive rate was only 18.9%. There was only 1 extreme event that was not predicted by our scheme, which means that the false negative rate was less than 1%. Furthermore, the ROPE method has spatial skill, correctly predicts the spatial location where the rogue wave will occur.

Figure 4: Probability of upcoming rogue wave as function of space and time (left) and value of wave envelope (right). Note that the predicted probability of upcoming rogue wave becomes large before the rogue wave actually occurs and gives the correct spatial location.

Our ROPE scheme for prediction of rogue waves reliably predicts rogue waves before they occur in a variety of one-dimensional nonlinear dispersive models. However, in order to work in realistic ocean settings, the method must be robust enough to make predictions using noisy, incomplete two-dimensional data. We are currently testing ROPE with this type of input. Our target is to be able to provide an early warning system where rogue/large waves are predicted 30 seconds to a few minutes in advance. Such a scheme would provide the crew onboard a vessel some time to prepare for the impact of the upcoming wave.

Here we study dynamical systems that are subjected to intermittent (rare) instabilities due to the presence of external stochastic excitation; in other words systems where rare, but extreme responses are sporadically observed. Systems of this kind are found in turbulent fluid flows and nonlinear water waves where nonlinear energy exchanges often occur in an intermittent fashion. Another class of systems where intermittency is observed is mechanical configurations subjected to parametric excitations, such as parametric resonance of oscillators and ship rolling motion. In addition, extreme rare responses are typical in nonlinear systems that contain an invariant manifold that locally loses its transverse stability properties, e.g. in the motion of finite-size particles in fluids, but also in biological and mechanical systems with slow-fast dynamics. In Figure 5 we observe a rare event in the response (lower plot) of a parametrically excited oscillator by correlated noise (upper plot). The result of intermittent instabilities can be observed in the heavy tails of the probability density function for the system response.

Figure 5: Typical response for a system that experiences intermittent instabilities due to stochastic excitation: a toy model for ship rolling motion under irregular waves. The system becomes unstable when the stochastic parameter alpha (upper plot) becomes negative. When this happens the system experiences an extreme response (lower plot).

The overall goal of this work is to develop an analytical method for the description of the system's response (the system response's probability distribution function), for the case when intermittent instabilities and extreme responses occur due to parametric excitation by correlated (colored) noise. The presence of colored noise makes the analysis particularly challenging, with no exact analytical techniques at our disposal. However, from a practical standpoint taking into account the correlation in the stochastic excitation process is critical, since most signals in nature are described by colored stochastic process and this correlation significantly influences the form of the non-Gaussian tails of the distribution. Overall, we seek a simple analytical method that can approximate the exact probability function of the system response, since numerical simulations are computationally expensive (many simulations must be computed to capture the statistics of rare events); moreover, analytical results help aid and enhance our understating of the underlying physics and the development of design criteria.

The technique we employ relies on decomposing the problem and separately analyzing the two regimes of the system (stable and unstable states) [M. Mohammad, T. Sapsis, 2015]. We rely on Bayes' rule, which allows us to condition the full probability density function of the response on the two regimes. Instead of solely analyzing the statistics of the heavy-tails or the system dynamics, our method utilizes both the statistics and the system dynamics to arrive at an accurate analytical approximation to the system probability distribution function.

Figure 6: Example of a typical decomposition of the full solution into the unstable part and stable part.

A direct application of our results is the analytical description of parametrically excited mechanical systems such as ship rolling motion in irregular ocean waves, known as parametric rolling. Under rare situations, the right wave can cause the ship to undergo large amplitude roll motions. The developed method allows us to take into account the correlation in the stochastic excitation resulting in a probability distribution function that matches the heavy-tails observed in direct numerical simulations of the corresponding models. In practice, it is critical to have a simple analytical description of the probability of roll motion for a variety of applications, including insurance models and risk assessment (especially important for large cargo ships, where loss of cargo containers is a major concern), safety specifications in ship hull form, the design of control systems, and rare event prediction. Current research efforts of our group focus on the application of the developed methodology to realistic ship roll models.

We also apply our results to the analytical description of the probability distribution function of modes in turbulent signals. Describing intermittency in these settings is an essential aspect for the development of efficient uncertainty quantification and filtering methods. Our analytical approximations can accurately capture the form of the probability density function in various regimes of the dynamics ranging from a very turbulent regime characterized by frequent instabilities to a regime where instabilities are extremely rare and in between.

Figure 7: An intermittent signal that represents a complex mode of a turbulent field (left). Comparison between the developed analytical method and the pdf generated by direct numerical simulations (right).